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This is a question on cap volatility market data. The quotes usually include volatilities for different strike (1%, 2%, ... 5%) and maturities (1Y,2Y,...20Y). One volatility for each combination of strike and maturity.

If I want to price a cap with, let's say, a strike of 2% and maturity in 10 years I would use the corresponding volatility from the market data described above. I think I understand it so far.

But then I also have volatility quotes for the at-the-money (ATM) rate. How do I know what the current at-the-money rate is? Second, in which situation would I use the ATM quotes?

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    $\begingroup$ To find the ATM rate you have to look at swaps of the corresponding maturity. $\endgroup$ – noob2 Nov 8 '17 at 20:55
  • $\begingroup$ Just trying to rephrase your answer to make sure I understand it: When I have a 10y cap and the cap strike is by incident the 10y swap rate, then I would use the ATM-10Y cap volatility for pricing. Is that right? $\endgroup$ – rokeby Nov 8 '17 at 21:11
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    $\begingroup$ Yes. But it is not a rare event, a considerable amount of business gets done at that rate. $\endgroup$ – noob2 Nov 8 '17 at 21:21
  • $\begingroup$ And why is that? Why would I want to have a cap with this strike? $\endgroup$ – rokeby Nov 8 '17 at 22:29
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The question is 1 year-old old but I will answer it anyway. The ATM level (ATMF: at the money forward to be more precise) is the one giving you the same price for call and put, or in this case, the same price for cap and floor.

So, let us start with writing the cap / floor parity: $$Cap(K) - Floor(K) = Swap(K)$$ where $Swap(K)$ is a swap paying K and that has the exact same characteristics (maturity, schedule, etc.) as the cap and floor.

The ATM level is then the one for which $Swap(K) = 0$, and that is, by definition, the swap rate of this swap.

Now, why do people define strikes with respect to this ATM level. Roughly speaking, it's the level such that you have a 50/50 chance of ending above or bellow of it at expiry. So, it makes more sense to define a set of strikes relative to this ATM level rather than having absolute strikes that can be useless after a move of market parameters.

Another interesting property is that the ATM level is where the option's time value is maximal.

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